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### A Construction of Monotonically Convergent Sequences from Successive Approximations in Certain Banach Spaces.

Numerische Mathematik

### A convergence analysis of Newton-like methods for singular equations using outer or generalized inverses

Applicationes Mathematicae

The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...

### A convergence analysis of Newton's method under the gamma-condition in Banach spaces

Applicationes Mathematicae

We provide a local as well as a semilocal convergence analysis for Newton's method to approximate a locally unique solution of an equation in a Banach space setting. Using a combination of center-gamma with a gamma-condition, we obtain an upper bound on the inverses of the operators involved which can be more precise than those given in the elegant works by Smale, Wang, and Zhao and Wang. This observation leads (under the same or less computational cost) to a convergence analysis with the following...

### A Convergence Theorem for Newton-Like Methods in Banach Spaces.

Numerische Mathematik

### A convergent nonlinear splitting via orthogonal projection

Aplikace matematiky

We study the convergence of the iterations in a Hilbert space $V,{x}_{k+1}=W\left(P\right){x}_{k},W\left(P\right)z=w=T\left(Pw+\left(I-P\right)z\right)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W\left(P\right)$ is continuous and the Lipschitz constant $∥\left(I-P\right)W\left(P\right)∥<1$. If an operator $W\left({P}_{1}\right)$ satisfies these assumptions and ${P}_{2}$ is an orthogonal projection such that ${P}_{1}{P}_{2}={P}_{2}{P}_{1}={P}_{1}$, then the operator $W\left({P}_{2}\right)$ is defined and continuous in $V$ and satisfies $∥\left(I-{P}_{2}\right)W\left({P}_{2}\right)∥\le ∥\left(I-{P}_{1}\right)W\left({P}_{1}\right)∥$.

### A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.

### A general iterative approach to variational inequality problems and optimization problems.

Fixed Point Theory and Applications [electronic only]

### A general semilocal convergence result for Newton’s method under centered conditions for the second derivative

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein type.

### A general semilocal convergence result for Newton’s method under centered conditions for the second derivative

ESAIM: Mathematical Modelling and Numerical Analysis

From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear...

### A Generalisation of Régula Falsi.

Numerische Mathematik

### A generalization of Edelstein's theorem on fixed points and applications.

Southwest Journal of Pure and Applied Mathematics [electronic only]

### A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems.

Abstract and Applied Analysis

### A hybrid-extragradient scheme for system of equilibrium problems, nonexpansive mappings, and monotone mappings.

Fixed Point Theory and Applications [electronic only]

### A local convergence analysis and applications of Newton's method under weak assumptions.

Southwest Journal of Pure and Applied Mathematics [electronic only]

### A local convergence theorem for the inexact Newton method at singular points.

Southwest Journal of Pure and Applied Mathematics [electronic only]

### A Method for Finding Sharp Error Bounds for Newton's Method Under the Kantorovich Assumptions.

Numerische Mathematik

### A modification of the Kantorovich conditions for the secant method.

Southwest Journal of Pure and Applied Mathematics [electronic only]

### A monotone convergence theorem for Newton-like methods using hypotheses on divided differences of order two.

Southwest Journal of Pure and Applied Mathematics [electronic only]

### A multi-step iterative method for approximating fixed points of Presić-Kannan operators.

Acta Mathematica Universitatis Comenianae. New Series

### A new approach for finding weaker conditions for the convergence of Newton's method

Applicationes Mathematicae

The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in ,  we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can...

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